1. Field of the Invention
The present invention is directed to compensating for smoothing which occurs in the interpolation of data arising in the conversion of data from a first geometry to a second geometry, and specifically to a computed tomography apparatus with such compensation.
2. Description of the Prior Art
The attenuation values of x-rays are measured in a plurality of directions in computed tomography in order to calculate cross sectional images of a specific subject. This measurement is based on a geometry, for example that of a fan beam system or of a parallel beam system. The desired computer tomograms are acquired by a reconstruction algorithm which is in turn based on a certain geometry. When one wishes to work with a different geometry in the image reconstruction than was used in the measurement (data acquisition), then the measured attenuation values (data) must be correspondingly converted before the reconstruction. This occurs by interpolation.
A problem arises in deriving data P.sub.k,l in a different raster .alpha..sub.kl =(1-.delta..sub.k).DELTA..theta. with 1=-M, . . . , M from data F.sub.n in a raster .alpha..sub.n =n.DELTA..alpha. with n=-M, . . . , M. The value .delta..sub.k is composed of a whole-number part .delta.'.sub.k =int(.delta..sub.k) and a fractional part .epsilon..sub.k =.delta..sub.k -.delta.'.sub.k. This conversion thus occurs from one raster to a different raster having a different sampling interval and a different raster position by an interpolation h(.alpha.) according to the relationship ##EQU1##
For explanatory purposes, the conversion onto a raster .alpha..sub.k,l =(1-.delta..sub.k).DELTA..alpha. with the linear interpolation ##EQU2## shall be considered below. The exemplary conversion thus occurs from a first raster to another raster having the same sampling interval, but having a different scanning raster position.
A problem is that such interpolations lead to a smoothing of the data. This is expressed in a reduction of specific frequencies in the spectrum of the interpolated data P.sub.k,l dependant on the properties of the type of interpolation employed.